Utilizing Multivariate Adaptive Regression Splines (MARS) for Precise Estimation of Soil Compaction Parameters

Abstract

Traditional laboratory methods for estimating soil compaction parameters, such as the Proctor test, have been recognized as time-consuming and labor-intensive. Given the increasing need for the rapid and accurate estimation of soil compaction parameters for a range of geotechnical applications, the application of machine learning models offers a promising alternative. This study focuses on employing the multivariate adaptive regression splines (MARS) model algorithm, a machine learning method that presents a significant advantage over other models through generating human-understandable piecewise linear equations. The MARS model was trained and tested on a comprehensive dataset to predict essential soil compaction parameters, including optimum water content (w_opt) and maximum dry density (ρ_dmax). The performance of the model was evaluated using coefficient of determination (R²) and root mean square error (RMSE) values. Remarkably, the MARS models showed excellent predictive ability with high R² and low RMSE, MAE, and relative error values, indicating its robustness and reliability in predicting soil compaction parameters. Through rigorous five-fold cross-validation, the model’s predictions for w_opt returned an RMSE of 1.948%, an R² of 0.893, and an MAE of 1.498%. For ρ_dmax, the results showcased an RMSE of 0.064 Mg/m³, an R² of 0.899, and an MAE of 0.050 Mg/m³. When evaluated on unseen data, the model’s performance for w_opt prediction was marked with an MAE of 1.276%, RMSE of 1.577%, and R² of 0.948. Similarly, for ρ_dmax, the predictions were characterized by an MAE of 0.047 Mg/m³, RMSE of 0.062 Mg/m³, and R² of 0.919. The results also indicated that the MARS model outperformed previously developed machine learning models, suggesting its potential to replace conventional testing methods. The successful application of the MARS model could revolutionize the geotechnical field through providing quick and reliable predictions of soil compaction parameters, improving efficiency for construction projects. Lastly, a variable importance analysis was performed on the model to assess how input variables affect its outcomes. It was found that fine content (C_f) and plastic limit (PL) have the greatest impact on compaction parameters.

1. Introduction

Soil compaction, a foundational process in geotechnical engineering, refers to the method of tightening soil particles and reducing air voids, thereby increasing density while maintaining the existing water content [1]. This technique not only enhances soil’s mechanical behavior but also provides a stable base for various infrastructures, solidifying its significance in the field of civil engineering. Initially suggested by Proctor in 1933 [2], soil compaction involves applying specified compactive effort in a controlled laboratory setting, allowing for various water contents. This process enables engineers to pinpoint the optimal moisture content and maximum dry density—two essential parameters obtained from the compaction curve [3,4,5]. In field compaction, the key parameters are the optimum water content (w_opt) and maximum dry density (ρ_dmax). These are crucial in determining the maximum dry density of soil, which indicates an enhanced soil bearing capacity. The optimum water content is the specific moisture level at which soil can attain its highest dry density following a certain compaction effort. It is the minimum water necessary to form a thin film on soil particles, aiding their sliding movement for compaction. Compressing soil to its greatest theoretical density involves eliminating gases within the soil via expelling them through voids in the soil matrix. This process of gas expulsion is fundamental to soil densification, allowing soil to reach its maximum potential density. Hence, w_opt and ρ_dmax are vital in achieving the desired soil densification and stability [6]. These parameters form the backbone of long-term performance for a myriad of geotechnical structures, such as landfill liners, highway embankments, railway track-beds, earth dams, nuclear waste disposal facilities, and airfield pavements [7,8,9,10,11,12,13,14,15,16,17]. For example, study [7] found that w_opt and ρ_dmax are crucial factors affecting the compression of compacted Oklahoma soils, influencing collapse potential in embankments. Another study [8] used w_opt and ρ_dmax as critical parameters to understand the behavior and performance of MX80, which is a versatile clay for nuclear waste barriers. Furthermore, study [14] investigated the microstructure and hydraulic properties of high-speed railway subgrade soil using different values of w_opt and ρ_dmax to understand its behavior. Thus, accurately predicting and understanding the compaction parameters of diverse soil types is paramount for the successful construction and maintenance of these geotechnical structures.

To find compaction parameters, two main compaction approaches emerged: the standard Proctor compaction and the modified Proctor compaction [1]. The standard method is typically adopted for regular traffic loading situations, while the modified approach is used when dealing with heavy unit weights like those considered for airfield pavements. Outcomes derived from laboratory tests conducted using either the standard Proctor compaction method (ASTM D698) [18] or the modified Proctor compaction method (ASTM D1557) [19] are visually represented via an inverted ‘V’ curve. This graphical representation illustrates the peak, known as ρ_dmax, and the corresponding moisture content, termed as the w_opt of the soil. Reaching the optimum moisture content and maximum dry density in a laboratory setting, using both standard and modified compaction tests, demands significant resources. It involves a considerable amount of time (typically 2–3 days), substantial effort, and a large volume of soil (around 20 kg per individual test) [1]. Efforts have been made to overcome this challenge through proposing several prediction models for determining the soil compaction parameters more efficiently. Empirical equations, created via multiple linear regression, can predict w_opt and ρ_dmax of soil [20,21,22,23,24,25]. These models offer the benefits of quick calculations and easy application, which speed up decision-making in geotechnical projects. They are user-friendly, making them accessible to a variety of users, not just experts. However, these models come with several limitations. One issue is the existence of numerous equations for the same factors, leading to potential confusion and inconsistency in results. This vast array of options makes the selection of the appropriate model challenging, possibly affecting the reliability of predictions. Another significant limitation is the assumption of linear behavior, which often does not reflect the complex, often non-linear nature of soil behavior. These models, while providing an estimation, might not fully encapsulate the range of outcomes in different conditions due to their simplified assumptions.

The complex nature of soil compaction parameters, influenced by numerous interrelated parameters, necessitates the use of robust, data-driven methodologies such as artificial intelligence. Traditional models struggle to capture these intricate relationships, but artificial intelligence and other data-driven methods have shown proficiency in navigating the inherent non-linearity and successfully estimating soil compaction parameters. Continued research should focus on employing precise and reliable techniques, like artificial intelligence, that can effectively handle these complexities. In the area of predicting soil compaction parameters, several studies have demonstrated noteworthy advancements. The use of artificial neural networks (ANNs) has been successfully employed in various models. For instance, one investigation applied an ANN to model w_opt and ρ_dmax in soil stabilized with nanomaterials. The model exhibited impressive prediction accuracy of over 97%, surpassing conventional statistical models [26]. Extending this approach to different soil types, another study harnessed the potential of deep neural networks to predict soil compaction parameters specific to Egyptian soil. The study achieved remarkable accuracy, with R² values of 0.864 for ρ_dmax and 0.924 for w_opt, thereby surpassing the performance of a simple artificial neural network with just one hidden layer [27]. In the context of highway projects, a multi-layer perceptron neural network model was developed to predict modified compaction parameters for both coarse and fine-grained soil samples. This model, similar to its predecessors, delivered highly accurate results [28]. Stepping beyond traditional ANNs, a novel hybrid intelligence paradigm known as the ANFIS-IGWO model was proposed in a different study. This model demonstrated superior precision in predicting soil compaction parameters, boasting correlation values of 0.9203 and 0.9050 for w_opt and ρ_dmax, respectively [29]. This accuracy was found to surpass other hybrid ANFIS models. In yet another study, an ANN was utilized to develop predictive equations for the Proctor compaction parameters of fine-grained soil [30]. The resulting ANN model demonstrated high efficiency and accuracy, particularly in predicting unseen datasets, thus setting a new standard compared to existing models in the literature. Moving towards the integration of empirical correlations with machine learning algorithms, one paper investigated the use of the support vector machine algorithm to predict soil compaction properties [31]. This approach aimed at significantly reducing the time and effort required in the laboratory, achieving R² values of 0.86 and 0.91 for w_opt and ρ_dmax, respectively. Lastly, in a significant stride towards machine learning models, a novel model called “ComPara2021” was proposed [17]. This model, based on machine learning, demonstrated superior performance in predicting soil compaction parameters compared to its contemporaries, proving the value of such models in saving both time and cost.

Another form of artificial intelligence is genetic programming approaches such as gene expression programming (GEP) and multigene expression programming (MEP). GEP and MEP are advantageous frameworks that integrate the benefits of both genetic programming and genetic algorithms, enabling them to evaluate more complex functions that express the relationship between input and output data [32,33]. The GEP and MEP models have notably demonstrated their effectiveness in predicting a variety of parameters related to compacted soil [15,34,35].

The multivariate adaptive regression splines (MARS) technique, a form of artificial intelligence, offers potential benefits to geotechnical engineers. Besides being adept at tackling prediction problems, MARS can identify key input parameters that significantly influence output parameters. It also allows for the exploration of intricate nonlinear relationships between a response variable and various predictive variables—a crucial aspect in analyzing and designing soil compaction parameters. This technique has seen successful implementations in diverse fields of geotechnical engineering, including ultimate pile bearing capacity [36], the elastic modulus of rocks [37], Slope reliability analysis [38], the compressive strength of soil [39], liquefaction [40], settlement prediction [41], and penetration resistance in clay [42]. Yet, despite its proven effectiveness in these areas, its application in compaction-related studies remains surprisingly limited.

In this research, MARS is utilized to estimate the optimal moisture content (w_opt) and maximum dry density (ρ_dmax) of compacted soil. MARS addresses the limitations of the multiple linear regression (MLR) method, as it does not necessitate an a priori form of equation between dependent and independent variables and can precisely capture complex and nonlinear relationships between inputs and outputs [43]. Unlike techniques like the ANN, which often operate as a “black box,” MARS can provide an explicit equation linking inputs and outputs [44,45,46]. This transparency has led to its frequent application in modeling intricate engineering issues. Finally, MARS successfully mitigates one of the key drawbacks of GEP and MEP, which tend to produce highly nonlinear mathematical relations between input and output data [46].

2. Research Significance

This research carries significant implications in the field of soil engineering and artificial intelligence applications. It provides a robust foundation for understanding and predicting soil compaction parameters across a variety of soil classifications. Firstly, this study is rooted in comprehensive literature-based data collection, which has amassed an extensive database of different soil types, including gravel, sand, silt, and clay. This broad spectrum of data offers a detailed and diverse basis for analysis and prediction. Secondly, we employed the MARS model to predict soil compaction parameters. To enhance the accuracy and reliability of predictions, we further fine-tuned the MARS model’s performance via hyperparameter tuning. Thirdly, we conducted a comparative study to evaluate the performance of the MARS model against other existing empirical and artificial intelligence models. This comparison seeks to identify the most effective and accurate model for predicting soil compaction parameters, contributing to the optimization of methodologies in this area of research. Lastly, we carried out a sensitivity analysis to determine the relative influence of each input variable on the scour depth. Through identifying the most impactful variables, this study aids in refining the prediction model and enhances the overall understanding of the relationships between different factors.

3. Materials and Methods

3.1. Research Methodology

The comprehensive research approach employed in this study is depicted in Figure 1, encompassing five fundamental phases: gathering and assimilating data, visualizing and statistically assessing the data, carrying out data modeling and analysis, evaluating the performance of machine learning (ML) models, and comparing the findings with earlier research. The data modeling and analysis are conducted using R-project, a programming framework extensively used for data mining and ML applications [47].

3.2. Multivariate Adaptive Regression Splines (MARS)

In the year 1991, the concept of multivariate adaptive regression splines (MARS), a method for non-parametric regression, was proposed by Friedman [48]. This approach aims to model a relationship between a set of predictor variables, represented as X (n × p), and a target variable, denoted as y (n × 1). Where n is the number of observations or samples, p is the number of predictor variables or features, X is the matrix of predictor variables with size n × p, and y is the column vector of the target variable with size n × 1. The MARS model’s mathematical representation can be expressed through the equation below:

$$y=f\left(X\right)+e$$

(1)

The term f(X) represents the prediction or approximation of the target variable y given the predictor variables X by the MARS model. The equation also gives us a residual vector, e (n × 1), which forms an integral part of this model. MARS can be considered as a sophisticated extension of the Classification and Regression Trees (CART) [49] technique, but it stands apart from conventional parametric methods. While parametric methods necessitate a predefined functional correlation between predictor and target variables, MARS refrains from such presumptions. Instead, MARS utilizes the predictor variables present in the specified dataset to produce a set of coefficients and piece-wise polynomials that carry the power $‘q’$. These polynomials are specifically generated to encapsulate the relationship among variables, which may exhibit either a linear or a non-linear nature. These polynomials, referred to as ‘splines’, are seamlessly connected to assemble the complete MARS model. Moreover, the process of fitting these splines involves segmenting the data points of the independent variables into discrete regions, with the knot locations symbolized by ‘$t$’. The power ‘$q$’ of these splines essentially determines their linearity or non-linearity. For any variable ‘$x$’, the MARS model calculates these splines through applying the particular equations below:

$${[-(x-t\left)\right]}_{+}^q=\left\{\begin{array}{ll}{(t-x)}^q,& \hspace{1em}\mathrm{if}\hspace{0.17em}x<t\\ 0,& \hspace{1em}\mathrm{otherwise}\end{array}\right.$$

(2)

$${[+(x-t\left)\right]}_{+}^q=\left\{\begin{array}{ll}(t-x{)}^q,& \hspace{1em}\mathrm{if}\hspace{0.17em}x\ge t\\ 0,& \hspace{1em}\mathrm{otherwise}\end{array}\right.$$

(3)

In the given equations, ‘$q$’ is not less than zero, and it is a key factor in shaping the smoothness of the ultimate MARS model. In this particular investigation, ‘$q$’ is set to 1, signifying the analysis of linear splines. Figure 2 illustrates two splines related to an individual variable ‘$x$’, having ‘$q$’ set to 1 and a defined knot ‘$t$’. The left spline provides positive outcomes, while the right spline produces zero for ‘$x$’ values on the left side of the knot. On the flip side, when ‘$x$’ values lie on the right side of the knot, the right spline gives positive results, and the left spline is zero. These splines, also called as the foundation functions for ‘$x$’, play a significant role in the MARS technique. The MARS model related to the aim variable ‘y’ is formed via including a series of ‘M’ basis functions, as demonstrated in the following equation:

$$\widehat{y}={\widehat{f}}_M(x)=a_o+{\sum }_{m=1}^M{a}_m{B}_m(x)$$

(4)

Equation (4) depicts $\widehat{y}$ as the target variable, which the MARS model predicts. The term ‘$a_o$’ signifies the intercept term, which is a fixed constant. ‘$B_m\left(x\right)$’ denotes the mth basis function, which could be an individual basis function or a combination of multiple basis functions, while ‘$a_m$’ symbolizes the coefficient that corresponds to the mth basis function. The estimation of these coefficients is achieved via the least squares method.

The MARS model, as represented by Equation (4), is formulated using a biphasic methodology that encompasses a forward pass and a backward pass. The initial phase, known as the forward pass, involves integrating the predictor variables into the model and fine-tuning the positions of their corresponding knots. This process gives rise to the creation of double-sided basis functions, one for each predictor. In the context of an X matrix, composed of ‘n’ samples and ‘p’ predictors, ‘$n$ × $p$’ pairs of basis functions emerge, as exhibited in Equations (2) and (3). These functions are associated with knots, denoted by $x_{ij}$, where ‘$i$’ ranges from 1 to ‘n’ and ‘$j$’ spans from 1 to ‘$p$’. Every single data point affiliated with a predictor variable is taken into account as a possible knot for that variable’s pair of basis functions. However, the forward pass process often results in the formulation of a convoluted model that tends to overfit, hence reducing its predictive capability. To address this issue, the second phase, or the backward pass, is implemented. During this phase, the model is systematically simplified through sequentially eliminating the least contributing, and therefore redundant, basis functions. The backward pass continues until an optimized sub-model is achieved, characterized by its lowest Generalized Cross Validation (GCV). This enhancement substantially boosts the predictive prowess of the MARS model. The computation for determining the GCV follows the formula provided below:

$$\begin{array}{c}GCV\left(M\right)=\frac{\frac{1}{n}{\sum }_{i=1}^n\hspace{0.17em}{\left(y_i-f_M\left({\widehat{x}}_i\right)\right)}^2}{{\left(1-\frac{M+d\times \left(M-1\right)/2}{n}\right)}^2}\end{array}$$

(5)

3.3. Hyperparameter Tuning Procedure

In this research, we utilized a combination of hyperparameter tuning, taking advantage of the grid search approach [50], along with cross-validation (CV) strategies. The grid search methodology essentially provides an ideal setting for parameters through arranging all variable grids within the space of the parameters. Each grid’s axis corresponds to a parameter of the algorithm, with a unique set of parameters at each grid point that requires optimization. The primary benefit of combining grid search with CV is to pinpoint the most suitable hyperparameters for the model based on the selected evaluation metric. Furthermore, this integration guarantees that the model’s performance is evaluated across various data subsets, resulting in a more robust and dependable assessment of its efficacy. A potential drawback of this methodology is that it is computationally intensive. The model needs to be trained ‘k’ times, which can be time-consuming for large datasets or complex models. However, since our data only has 226 points, the computational demand in our specific case is considerably reduced, making the method more manageable and feasible for our dataset size. To ensure the unbiased selection of data, we employed a prevalent validation method known as k-fold CV [51,52,53,54,55] during the tuning process of hyperparameters for MARS models. K-fold CV, a standard type of CV, is extensively used in machine learning. Although there is no definitive rule for choosing the value of K, in practical machine learning scenarios, K is commonly set at 5 or 10. According to Rodriguez [56], the bias in an accurate estimation is likely to be reduced when the fold count is either 5 or 10. Based on this and the suggestions of Kohavi [57] and Wong [58], we chose 5 for K. This decision also took into consideration the trade-off between computational efficiency and bias. As a result, five different sets of training and validation runs were performed, with the outcomes being averaged to depict the overall performance of the MARS models on the training dataset. All the data-related operations in this study were handled using R software 4.1.3 (R Core Team, Vienna, Austria) [47]. The approach for hyperparameter tuning used in this research for training the model and selecting hyperparameters is depicted in Figure 3, as illustrated in reference [59].

Our research made use of the R package “earth” to perform MARS modeling. This implementation called for the user’s careful consideration of two particular tuning parameters: “degree” and “nprune”. The “degree” parameter indicates the highest degree of interaction allowed in the model. By default, this value is set to 1, leading to the creation of a MARS model that is additive and lacks interaction terms. However, it is noteworthy that MARS is capable of building models that encompass interactions of a degree of two or even higher. Despite this capability, a guideline suggested by Hastie et al. [60] recommends maintaining an upper limit on the degree of interaction. The benefit of keeping a lower degree of interaction lies in its ability to assist in interpreting the final model in a more straightforward manner. On the other hand, a higher degree of interaction could potentially cause occasional volatility in the model’s predictive capabilities, possibly causing predictions to deviate significantly from actual values. Hence, to evaluate the impact of varying degrees of interaction, we experimented with three different values: 1, 2, and 3. The second parameter, “nprune”, signifies the maximum number of terms (inclusive of the intercept) that can be present in the model. The “nprune” value should be equal to or greater than 2 and less than “nk”. Here, “nk” stands for the maximum number of terms in the model before any pruning activity is initiated. This value of “nk” is determined by the formula nk = min (200, max (20, 2 × ncol(x))) + 1, where ncol(x) is the total number of predictive variables. For our specific study, which involved a total of 6 predictive variables, we looked into 20 potential values for nprune, ranging from 2 to 20. The process of constructing and evaluating these 57 MARS models, featuring degrees of 1, 2, 3 and nprune values ranging from 2 to 20, was accomplished through the implementation of a five-fold cross-validation procedure.

3.4. Performance Metrics

Three error evaluation metrics, namely root mean squared error (RMSE), mean absolute error (MAE), and the coefficient of determination (R²), are chosen to assess the accuracy of MARS model predictions. The R² metric represents the linear relationship between the predicted and actual values, with its value ranging from zero to one. A value of R² nearing one signifies a more accurate model prediction. RMSE provides insight into the dispersion between predicted and real data and is sensitive due to its relation to the Euclidean distance. Meanwhile, MAE is a measure of the average magnitude of errors between predicted and observed values. Lower values of RMSE and MAE suggest superior prediction performance by the algorithm. The three metrics are outlined as follows:

Root mean squared error (RMSE):

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\frac{{\sum }_{i=1}^n{\left(G_{pi}-G_{Ti}\right)}^2}{n}}$$

(6)

Correlation of determination (R²):

$${\mathrm{R}}^2=1-\frac{{\sum }_{i=1}^n{\left(G_{pi}-G_{Ti}\right)}^2}{{\sum }_{i=1}^n{\left({\bar{G}}_{Ti}-G_{Ti}\right)}^2}$$

(7)

Mean absolute error (MAE):

$$\mathrm{M}\mathrm{A}\mathrm{E}=\frac{1}{n}\sum _{i=1}^n\hspace{0.17em}\left|G_{pi}-G_{Ti}\right|$$

(8)

where $G_{Ti}$ and $G_{pi}$ are the true and predicted values for the observation, respectively. ${\bar{G}}_{Ti}$ is the mean of all the true values. $n$ is the number of samples.

4. Database Used

In the current research, we utilized data (comprising 226 entries) sourced from the study by [61], which span various soil categories such as fat clay (CH), silty clay (CL-ML), lean clay (CL), clayey gravel (GC), silty gravel (GM), elastic silt (MH), well-graded sand with clay (SW-SC), poorly graded sand with clay (SP-SC), silt (ML), clayey sand (SC), silty sand (SM), poorly graded gravel with clay (GP-GC), and well-graded gravel with clay (GW-GC). The physical attributes of these soil samples are presented in

Table 1. Model variable descriptive statistics.

Statistics	CG (%)	CS (%)	CF (%)	LL (%)	PL (%)	E (kJ/m3)	wopt (%)	ρdmax (Mg/m3)
Standard deviation	14.566	23.388	30.016	164.218	7.405	735.435	5.964	0.199
Mean	7.468	29.448	63.088	108.727	22.005	894.066	17.512	1.751
Median	0.000	27.000	70.000	40.650	20.150	593.000	17.000	1.750
Maximum	67.100	89.000	100.000	608.000	48.300	2755.000	43.700	2.330
Minimum	0.000	0.000	8.600	16.000	6.100	155.000	5.300	1.090
Kurtosis	3.482	−0.471	−1.250	3.264	0.748	2.266	2.862	0.977

. This dataset encompasses a diverse array of soils, highlighted by the extensive variety of index properties. Within this dataset, it is posited that the compaction metrics are influenced by factors like liquid limit (LL), plastic limit (PL), compaction energy (E), sand content (C_S), fines content (C_F), gravel content (C_G), optimum water content w_opt, and maximum dry density ρ_dmax, all of which are detailed in Table 1. Figure 4 presents frequency histograms for each variable, inclusive of soil classification. Analyzing Figure 4 reveals that within this database, a significant portion of soils possess gravel and sand contents ranging from 0% to 20% (as seen in Figure 4a,b). Conversely, the content of fines in the soils predominantly falls within 80% to 100% (illustrated in Figure 4c). A large portion of soils register a liquid limit between 0 and 50 (as displayed in Figure 4g), and 32 soils exhibit a liquid limit exceeding 300, indicative of high plasticity. The plastic limit is chiefly found between 10 and 30 (Figure 4d). Notably, this dataset features a comparably larger quantity of clayey sand, fat clay, lean clay, and clayey gravel in contrast to other soil types (depicted in Figure 4f). A significant number of compaction tests were executed under standard Proctor conditions or with diminished compaction energy (not surpassing 600 kJ/m³ as shown in Figure 4e). Around 30 modified Proctor compaction tests are also part of the study. In terms of maximum dry density and optimum water content, peak frequencies are predominantly within 1.6 Mg/m³ to 1.8 Mg/m³ and 10% to 20%, respectively (as captured in Figure 4h,i).

5. Model Results

5.1. Hyperparameter Tuning Results for Optimal Model

The hyperparameter tuning procedure, which combined a grid search and k-fold cross-validation, was employed to determine the optimal model parameters for the MARS models, aiming for the lowest RMSE. According to Figure 5, the performance of the MARS model was more sensitive to the ‘nprune’ parameter than the interaction degree, highlighting the crucial role of ‘nprune’ in achieving top-tier model accuracy.

As detailed in Figure 5, a rigorous five-fold cross-validation process pinpointed ‘degree = 2’ and ‘nprune = 13’ as the prime settings for the w_opt MARS model. Leveraging these parameters, the model showcased exemplary performance across the 131 training datasets, registering an average RMSE. On the other hand, for the ρ_dmax MARS model, the most suitable hyperparameters were ‘nprune = 11’ and ‘degree = 2’. With these settings, the model attained an optimal predictive capacity, reflected by an average RMSE of 0.064 and an R² of 0.889, specifically for predicting the maximum dry density in compacted soil.

5.2. Cross-Validation Results after Hyperparameter Tuning

Post tuning, the performance of the final MARS models was assessed using a five-fold cross-validation procedure. The RMSE, R², and MAE were computed for each fold to estimate the prediction performance. The results for both models are presented below.

For the w_opt model, the five-fold cross-validation results are presented in Table 2:

Table 2. Results derived from a five-fold cross-validation for the w_opt model.

Folds	Performance Measures
	RMSE (%)	R2	MAE (%)
Fold 1	1.513	0.929	1.182
Fold 2	2.368	0.870	1.867
Fold 3	1.827	0.886	1.351
Fold 4	1.710	0.889	1.361
Fold 5	2.323	0.895	1.728
Average	1.948	0.893	1.498
SD	0.380	0.023	0.287
CoV (%)	0.195	0.026	0.192

Table 2. Results derived from a five-fold cross-validation for the w_opt model.

Folds	Performance Measures
	RMSE (%)	R2	MAE (%)
Fold 1	1.513	0.929	1.182
Fold 2	2.368	0.870	1.867
Fold 3	1.827	0.886	1.351
Fold 4	1.710	0.889	1.361
Fold 5	2.323	0.895	1.728
Average	1.948	0.893	1.498
SD	0.380	0.023	0.287
CoV (%)	0.195	0.026	0.192

Notably, the RMSE, R², and MAE values indicate the high predictive performance of the MARS model for the w_opt parameter, with high R² values over 0.87 for all folds in both cases. This highlights the models’ ability to capture a significant portion of the variance in the target variables. The RMSE values were relatively low across all folds for both models, underscoring the models’ robustness and their ability to minimize errors in prediction. Moreover, the MAE values were reasonably low, revealing the models’ capability to produce accurate predictions with small deviations from the actual values.

The ρ_dmax model’s cross-validation results are presented in

Table 3. Results derived from a five-fold cross-validation for the ρ_dmax model.

Folds	Performance Measures
	RMSE (Mg/m3)	R2	MAE (Mg/m3)
Fold 1	0.068	0.860	0.051
Fold 2	0.067	0.908	0.050
Fold 3	0.048	0.948	0.040
Fold 4	0.071	0.870	0.057
Fold 5	0.065	0.909	0.049
Average	0.064	0.899	0.050
SD	0.009	0.035	0.006
CoV (%)	0.141	0.039	0.120

.

Once again, the RMSE, R², and MAE values for each fold reflect the high predictive performance of the model. The average values highlight the overall accuracy of the model, while the standard deviations suggest that the model’s performance is consistent across different folds.

One potential reason some folds perform better than others is due to imbalanced data distribution. If the data are not evenly distributed and one fold ends up with a disproportionate number of certain classes or types of data, the model may perform poorly on that fold. For instance, if one fold contains many outliers or rare events, the model might struggle to generalize. Despite this, the consistently high R² values, low RMSE and MAE values, and the low standard deviations for both models across all folds underscore the effectiveness of the MARS models and their robustness in predicting the w_opt and ρ_dmax parameters of compacted soil.

5.3. Evaluation of MARS Models on Unseen Dataset

The following two equations were derived from the final optimal models after hyperparameter tuning for w_opt and ρ_dmax. The equations are as follows:

$$\begin{array}{ll}{w}_{opt}& =19.3773983\\ & -0.0639987\times \mathrm{m}\mathrm{a}\mathrm{x}(97-CF,0)\\ & -5.1538333\times \mathrm{m}\mathrm{a}\mathrm{x}(CF-\mathrm{97,0})\\ & -0.0898632\times \mathrm{m}\mathrm{a}\mathrm{x}(64-LL,0)\\ & -0.6415284\times \mathrm{m}\mathrm{a}\mathrm{x}(21-PL,0)\\ & +0.4288445\times \mathrm{m}\mathrm{a}\mathrm{x}(PL-\mathrm{21,0})\\ & +0.0025993\times \mathrm{m}\mathrm{a}\mathrm{x}(1347-\mathrm{E}\mathrm{k}\mathrm{J}/{\mathrm{m}}^3,0)\\ & +0.2425873\times \mathrm{m}\mathrm{a}\mathrm{x}(CF-\mathrm{97,0})\times PL\\ & -0.0000061\times LL\times \mathrm{m}\mathrm{a}\mathrm{x}(\mathrm{E}\mathrm{k}\mathrm{J}/{\mathrm{m}}^3-\mathrm{1347,0})\\ & +0.0578447\times \mathrm{m}\mathrm{a}\mathrm{x}(7.6-CG,0)\times \mathrm{m}\mathrm{a}\mathrm{x}(21-PL,0)\\ & -0.0093695\times \mathrm{m}\mathrm{a}\mathrm{x}(70-CF,0)\times \mathrm{m}\mathrm{a}\mathrm{x}(PL-\mathrm{21,0}).\end{array}$$

(9)

$$\begin{array}{ll}{\rho }_{dmax}& =1.39385529\\ & +0.00261487\times \mathrm{m}\mathrm{a}\mathrm{x}(33.1-CS,0)\\ & +0.00621111\times \mathrm{m}\mathrm{a}\mathrm{x}(98-CF,0)\\ & +0.00505340\times \mathrm{m}\mathrm{a}\mathrm{x}(LL-\mathrm{64,0})\\ & +0.00337354\times \mathrm{m}\mathrm{a}\mathrm{x}(130-LL,0)\\ & -0.00526625\times \mathrm{m}\mathrm{a}\mathrm{x}(LL-\mathrm{130,0})\\ & -0.01028165\times \mathrm{m}\mathrm{a}\mathrm{x}(PL-\mathrm{11,0})\\ & -0.00019175\times \mathrm{m}\mathrm{a}\mathrm{x}(1347-\mathrm{E}\mathrm{k}\mathrm{J}/{\mathrm{m}}^3,0)\\ & -0.00188667\times \mathrm{m}\mathrm{a}\mathrm{x}(CF-\mathrm{98,0})\times PL\\ & -0.00005026\times \mathrm{m}\mathrm{a}\mathrm{x}(CS-\mathrm{53,0})\times \mathrm{m}\mathrm{a}\mathrm{x}(98-CF,0)\\ & +0.00002616\times \mathrm{m}\mathrm{a}\mathrm{x}(CF-\mathrm{30,0})\times \mathrm{m}\mathrm{a}\mathrm{x}(130-LL,0).\end{array}$$

(10)

The cross plots presented in the subplots (a) and (b) of Figure 6 portray the application of two previous MARS-based models in predicting two crucial parameters for compacted soil: the optimum water content (w_opt) and maximum dry density (ρ_dmax). These subplots showcase the comparison between the models’ predictions and the actual values of w_opt and ρ_dmax for the testing datasets. The plots illustrate that a significant accumulation of data points clustering near the Y = X line signifies the strength and reliability of the established models. Clearly, the implemented models yield predictions that align remarkably well with the unit slope line. As a result, the proposed models can confidently be considered highly reliable for estimating the w_opt and ρ_dmax of compacted soil throughout the testing and training phases.

Figure 7 showcases the relative error of the MARS-based models compared to the actual measurements of w_opt and ρ_dmax. The subplots in Figure 7 demonstrate that the outputs of the proposed MARS-based models exhibit an acceptable deviation from the real measured data. A significant portion of the data falls within the range of −10% to +10% for w_opt and −5% to 5% for ρ_dmax.

Additionally, the statistical evaluation of the proposed models is presented in

Table 4. Performance metrics of the MARS models for training, testing, and overall datasets.

		Training	Testing	Overall
wopt	MAE (%)	1.12	1.276	1.15
	RMSE (%)	1.392	1.577	1.428
	R2	0.942	0.948	0.943
ρdmax	MAE (Mg/m3)	0.04	0.047	0.041
	RMSE (Mg/m3)	0.05	0.062	0.052
	R2	0.936	0.919	0.931

, providing a comprehensive summary of the assessment. The table includes the MAE, R², and RMSE values for the training, testing, and overall datasets. The statistical evaluation clearly indicates that both MARS-based models exhibit outstanding prediction performance. The overall RMSE values for w_opt and ρ_dmax are 1.428 and 0.052, respectively, highlighting the models’ accuracy. The combined findings from Figure 6 and Figure 7, along with the statistical results in Table 4, demonstrate the effectiveness of our newly proposed correlations. They are capable of reliably estimating w_opt and ρ_dmax for compacted soil across various soil properties and operational conditions, showcasing high integrity and efficiency.

5.4. Comparison between MARS Model with Previously Developed Models

In order to thoroughly assess the reliability of our newly proposed models, we extended the statistical evaluation to compare them with one of the best predictive correlations, namely Multi Expression Programming (MEP) [61]. The comparison is based on various statistical criteria and prediction performance, as presented in

Table 5. Performance metrics of the MEP models for training, testing, and overall datasets.

Compaction Parameters	Performance Measures	Overall
wopt	MAE (%)	1.3
	RMSE (%)	1.68
	R2	0.921
ρdmax	MAE (Mg/m3)	0.054
	RMSE (Mg/m3)	0.073
	R2	0.867

and visually represented in the bar plots of Figure 8. Analyzing Table 5 and Figure 8 reveals that, despite the satisfactory prediction performance of MEP, our suggested models outperform it in accurately estimating the values of w_opt and ρ_dmax for compacted soil. Furthermore, MEP frequently creates intricate nonlinear empirical models that can be difficult to handle [62]. In turn, the MARS algorithm offers some distinct benefits. Notably, it effectively captures the intricate interactions between independent and dependent variables. Moreover, it eliminates the need to exert extra effort to confirm any preliminary assumptions regarding their relationship. This advantage becomes increasingly vital as the complexity of the problem grows [63].

5.5. Variable Importance Analysis

A primary strength of the MARS model is its capability to discern if a particular input factor is vital for predicting output values or if it merely offers a slight enhancement to the model’s precision.

The varImp function in the caret package [64] produces a matrix that displays the relative importance of features within the model, utilizing two distinct methods to assess feature significance:

• The raw residual sum-of-squares (RSS) method proceeds in two phases. Initially, it assesses the RSS decrease for each subset, contrasting it with the preceding subset’s value. Subsequently, for every relevant feature, it accumulates these reductions across all subsets that incorporate that feature. Finally, the overall sum of these reductions is analyzed. Features leading to substantial RSS declines hold greater importance.
• The generalized cross-validation (GCV) method operates analogously to the RSS method, but it employs GCV in place of RSS. GCV assesses feature performance in subsets, pinpointing the most pivotal subset (with smaller GCV values being preferable).

Figure 9 displays the outcomes of the contributions and the selections made by different independent variables on the prediction performance of w_opt and ρ_dmax.

Figure 9a displays the impact of various independent variables on the ρ_dmax MARS model. The figure is sorted based on the decreasing influence of these variables. The top two influential factors are PL (%) and C_F (%), with RSS values of 100 and 55.01, and GCV values of 100 and 75, respectively. These statistics suggest that PL (%) holds the most significant influence on the ρ_dmax MARS model, followed closely by C_F (%). Other factors listed in decreasing order of influence are E (kJ/m³), LL (%), and C_S (%).

Figure 9b showcases the influence of different independent variables on the w_opt MARS model. The two most impactful factors in this table are C_F (%) and PL (%). C_F (%) takes the lead with RSS and GCV values of 100, indicating its paramount significance in the model. PL (%) follows next with RSS at 63.49 and GCV at 85.71. These results allow us to state that C_F (%) has the most considerable influence on the w_opt MARS model, but PL (%) also retains significant sway. Other listed factors, in order of influence, include E (kJ/m³), LL (%), and C_S (%).

To more deeply assess the reliability of the MARS models, a parametric analysis can be undertaken. This type of analysis examines the alignment of the model’s predictions with existing geotechnical knowledge, experimental data, and anticipated outcomes. This study scrutinized how the predicted soil parameters from the suggested models responded to hypothetical data, spanning the lower and upper bounds of the data used in model training. This approach entailed modifying a single input variable while maintaining others at their average. Synthetic data for this individual parameter were produced via raising it step by step. These inputs were then used in the prediction equation to determine soil parameters. This method was iteratively applied for different variables to gauge the model’s reaction to each input change.

Figure 10 illustrates how the predicted w_opt changes in relation to the C_F (%), the LL (%), the PL (%), and the E (kJ/m³). Notably, as the C_F (%), the LL (%), and the PL (%) rise, the forecasted w_opt also goes up. In contrast, when the E (kJ/m³) grows, the predicted value goes down. These consistent changes align with the patterns observed for the w_opt in relation to each soil property as seen in the figure, indicating that the proposed model is accurate.

Figure 11 displays the parametric analysis of the anticipated ρ_dmax in relation to each input variable. Contrary to the w_opt, the forecasted ρ_dmax goes down as the C_F (%), LL (%), or PL (%) rises and as the E (kJ/m³) reduces. This trend mirrors the monotonic observations from the actual database illustrated in the figure. Such a parallel between the parametric study and the database in predicting the ρ_dmax further confirms the accuracy of the model introduced.

The findings from the variable importance analysis are consistent with previously published literature. A respected study explored the influence of C_F on the mechanical response of CDG (completely decomposed granite) subjected to dynamic compaction grouting [65]. Through integrating varying proportions of kaolin clay with CDG, this study aimed to understand the implications of fines concentrations on its compactive behavior. The findings indicated that an increment in C_F correspondingly results in a diminished ρ_dmax, while concurrently elevating the w_opt. In another study, sand–kaolin blends were compressed to investigate the connection between ρ_dmax, C_F, and specific gravity (Gs) across different moisture levels, with C_F ranging from 20% to 70% [66]. A prominent linear correlation (R² = 0.99) was found between ρ_dmax and the C_F/Gs² ratio. For PL, a respected paper investigates which index properties best correlate with compaction traits. Incorporating both the study’s findings and the existing literature, it was determined that the plastic limit correlates strongly with compaction features, such as optimum moisture content and maximum dry unit weight, more effectively than either the liquid limit or plasticity index. For the plastic limit (PL), a notable study examined which index properties have the strongest correlation with compaction characteristics [23]. Through integrating the study’s results with the established literature, it was ascertained that the plastic limit shows a significant correlation with compaction attributes, particularly w_opt and ρ_dmax, surpassing the correlations seen with either the liquid limit or the plasticity index.

6. Conclusions

Determining compacted parameters in geotechnical projects is vital for ensuring soil stability, optimizing foundation designs, mitigating settlement issues, and guaranteeing the safety and longevity of structures built on or within the ground. For this reason, the prediction of those parameters is imperative for proactive planning, cost-effective execution, and minimizing project risks. This research aims to predict those parameters via employing the MARS (multivariate adaptive regression splines) model. Compared to other black-box machine learning models, the MARS model offers advantages such as flexibility in capturing non-linear relationships, automated interaction detection, fewer tuning parameters, and the capability to produce more interpretable results, making it particularly suitable for geotechnical applications where understanding the underlying behavior is as crucial as prediction accuracy.

The results obtained through this research indicate the following:

• Hyperparameter tuning of the MARS model significantly enhanced its predictive performance, reducing the error rate and refining the model’s adaptability to complex geotechnical data.
• The final optimal models for parameters w_opt and ρ_dmax were evaluated using five-fold cross-validation. The findings showed consistency and high accuracy across all folds, demonstrating the model’s robustness and reliability in estimating these critical parameters.
• Testing the models on unseen data (testing dataset) revealed that they maintained a commendable level of precision and generalization, further substantiating their efficacy for real-world applications in geotechnical projects.
• Comparing the MARS model with other state-of-the-art models, such as Multi Expression Programming (MEP), indicated that while both models exhibit strong predictive capabilities, MARS demonstrated faster convergence and better handling of non-linear relationships, offering a more efficient and robust alternative for geotechnical parameter prediction.
• Finally, the variable importance analysis revealed that the fine content (C_F) and plasticity limit (PL) are the most influential factors driving the predictive accuracy of the model, highlighting their pivotal role in determining the geotechnical properties under study.

It is worth mentioning that ML research often presents inherent challenges and limitations. In this study, a notable limitation is the data sample size of 226 points. The model formulated in this research delivers the anticipated accuracy when analogous input parameters are subsequently applied. However, errors may arise in the analysis if the same parameters are used but lie outside our established range. Gathering more experimental data in the future is essential to enhance the model’s generalization capabilities. Upcoming studies will explore the potential of advanced ML techniques, such as deep learning, in predicting w_opt and ρ_dmax values.